Banach’s Doctorate:A Case of MistakenIdentityDANUTA CIESIELSKA AND KRZYSZTOF CIESIELSKI

IIn the academic year 2020/2021 we celebrate thehundredth anniversary of Stefan Banach’s doctorate.Many books describe the curious way in which Banach

supposedly obtained his PhD. The story, which will bepresented in the sequel, is eye-catching indeed. But is ittrue? There are several misleading stories about Banachmaking the rounds; see [12]. Anyway, before we come tothe story about his PhD, let’s start from the very beginning.

Banach was born in Krakow in 1892. Sometimes it ismentioned that Banachwas not the surnameof anyone in hisfamily but that of a washerwoman, Katarzyna Banach, intowhose care he was given shortly after his birth. This is nottrue. Banach was the son of Katarzyna Banach and StefanGreczek. His parents were unmarried, but that was notunusual. At the end of the nineteenth century, about one inevery four children in Krakow was born out of wedlock.Greczek was serving in the Austro-Hungarian army, and hewould not have been allowed to marry without permissionfrom the military authorities. Such permission was denied.When the child was several months old, he was placed in thecare of a foster mother, a laundry owner named FranciszkaPłowa,who togetherwithher daughterMaria took careof theyoung Stefan until he reached the age of majority.

It is not difficult to find the source of this error. At a con-ference on functional analysis organized in September 1960,Hugo Steinhaus (1887–1972) presented a talk on Banach(whohaddied in 1945), andhis lecturewas later published inthe journal Wiadomosci Matematyczne [33]. Steinhaus’s

presentationwas brilliant, but his article also contained somefactual errors. Steinhaus claimed thatBanachhadbeen raisedby a washerwoman named Katarzyna Banach and confusedBanach’s biological mother with his foster mother. More-over, he gave the wrong date for Banach’s birth (20 March1892 instead of 30 March; see [12]).

Steinhaus played a crucial, and unusual, role for Banachat the beginning of his mathematical career. In 1916, duringan evening walk in Planty Park in Krakow, Steinhaus heardthe words “Lebesgue integral,” so he joined the two youngmen who were conversing. They turned out to be Banachand Otton Nikodym (1887–1974). Steinhaus wrote thatBanach and Nikodym told him that they “had a third pal,Wilkosz, on whom they heaped lavish praise” [32, 33].Witold Wilkosz (1891–1941) would frequently conversewith Banach and Nikodym, but he was not with them onthat evening. Steinhaus realized that Banach had a superbmathematical talent and began mentoring him. Steinhausused to say that his best mathematical discovery was StefanBanach.1 For details, see [7, 8, 20, 32]. Note that whileSteinhaus indeed discovered Banach, he discovered neitherNikodym nor Wilkosz, both of whom later became out-standing mathematicians.

It is not unusual in the history of science that an other-wise trustworthy witness misrepresents events. WacławSierpinski (1882–1969) knew both Banach and Steinhausvery well, and he worked closely with Nikodym.2 In [30],he provides a wildly distorted account of the famous

1On the occasion of the hundredth anniversary of that famous talk, a bench with the figures of Banach and Nikodym (designed by Stefan Dousa) was unveiled in the

Planty Garden in Krakow (see [7]). The following year, a picture of this bench appeared on the cover of this magazine (Mathematical Intelligencer 39:1).2Since we mentioned the cover of the Mathematical Intelligencer featuring the bench with Banach and Nikodym, we might mention as well that the cover of the

Mathematical Intelligencer 17:1 (1975) exhibits a poster with Sierpinski’s face formed by his space-filling curve, designed by Fritz Lott.

© 2021 The Author(s). This article is published with open access at Springerlink.com, Volume 43, Number 3, 2021 1

https://doi.org/10.1007/s00283-020-10033-x

meeting of Steinhaus and Banach in Planty Park in Krakowin 1916. He reports that on that day, one of the parkbenches was occupied by three mathematicians: Steinhaus(“then a docent at Lwow University”), Nikodym, and WitoldWilkosz. Sierpinski writes, “They were discussing somemathematical problem which had not been solved at thattime. Banach eavesdropped, introduced himself into theirconversation and offered his opinion as to how it could besolved.” As we know from Steinhaus’s firsthand accountmentioned above, it was the other way round. It wasSteinhaus who overheard the mathematical conversation,and Banach and Nikodym were unknown to him at thetime (nor did he know Wilkosz, for that matter). Steinhauswas a participant in the event, so his version of the story isundoubtedly correct. The only discrepancy in his accountsis that sometimes he writes that he joined the conversationon unexpectedly hearing the words “Lebesgue integral”[33], while at other times, the words were “Lebesgue mea-sure” [32]. He may have heard both. Note also that contraryto Sierpinski’s description, Steinhaus had no connectionwith Lvov University in 1916 and at the time, did not holdthe position of docent anywhere.

This meeting is also recalled in [21]. However, onlySteinhaus and Banach are mentioned there (although thename of Nikodym appears elsewhere in that book).

In 1920, Steinhaus was appointed to a chair at JanKazimierz University in Lvov.3 Thanks to Steinhaus’sefforts, Banach obtained an assistantship at the LvovPolytechnic University. Now we come to the central pointof our story: Banach’s doctorate.

The story goes that Banach could not be bothered withwriting a thesis, since he was interested mainly in solvingproblems not necessarily connected to a possible doctoraldissertation. After some time, the university authoritiesbecame impatient. It is said that another university assistant(instructed by Stanisław Ruziewicz) wrote down Banach’stheorems and proofs, and those notes were accepted as asuperb dissertation. However, an exam was also required,and Banach was unwilling to take it. So one day, Banachwas accosted in the corridor by a colleague, who asked himto join him in a meeting with some mathematicians whowere visiting the university in order to clarify certainmathematical details, since Banach would certainly be ableto answer their questions. Banach agreed and eagerlyanswered the questions, not realizing that he was beingexamined by a special commission that had arrived fromWarsaw for just this purpose. In some sources [11, 19, 20],this event is described as only a possible version of events.Nevertheless, in several (mainly Polish-language) books, itis presented as fact. There is even a book on the phobiasand fears of great Poles that devotes a whole chapter toBanach and this story, claiming to demonstrate that Banachwas unable to deal with his own psyche and phobias,although even this story presents Banach simply as some-one who did not consider the PhD a very importantacquisition.

This is a really attractive story. But is there any chancethat it is true? On the one hand, one may recall here afamous anecdote supposedly explaining the nonexistenceof the Nobel Prize in mathematics (see [4]): Niels Bohr usedto say that there is no Nobel Prize in mathematics becausethe Swedish mathematician Gosta Mittag-Leffler hadseduced Nobel’s wife. When Bohr was reminded that thiswas impossible, because Nobel had never married, Bohranswered, “Never let facts interfere with a good story.”Incidentally, the explanation for the lack of a Nobel Prize inmathematics given in [4] is also misleading; for an excellentdiscussion of this problem, see [17].

On the other hand, good stories aside, the truth aboutBanach’s exam should be known. Nowadays, it is possibleto check the facts, since many sources have become moreeasily available than they were some decades ago. It isenough to look carefully at some dates and university rulesto see that the proposed account could not be accurate.Banach moved to Lvov in 1920 to take up his job at theLvov Polytechnic. On June 24 of that year, he presented hisdoctoral dissertation to the Philosophy Faculty of JanKazimierz University. The time interval of just a couple ofmonths was definitely too short for the university authori-ties to have become impatient, let alone for someone elseto have written a thesis on the basis of Banach’s overheardcomments. Moreover, in 1920, Banach had already

Figure 1. Stefan Banach in 1919. (Courtesy of the Banach

family collection [19].)

3After World War I, Poland regained independence, and Lvov (now Lviv, Ukraine) reverted from Austria–Hungary to Poland. In 1919, the university was named after Jan

II Kazimierz Waza, the Polish king who chartered the university in Lvov in 1661.

2 THE MATHEMATICAL INTELLIGENCER

published three research papers. Why would he be reluc-tant to write a doctoral dissertation, which would be arequirement for him to keep the job?

Now let’s have a closer look at the exam. According tothe university rules, a PhD dissertation had to be refereedand accepted, and then two exams—in the candidate’smain scientific disciplines (in Banach’s case they weremathematics and physics) and in pure philosophy—wereto be taken by the candidate. It turns out that the records ofBanach’s PhD exams have survived (they are reproduced in[22] and [26]), and we may read that Banach passed his PhDexaminations in mathematics and physics. The examiningboard consisted of four scientists: the dean of the faculty,Zygmunt Weyberg, who was a mineralogist; two mathe-maticians, Eustachy Zylinski and Hugo Steinhaus; and aphysicist, Stanisław Loria. None of them was from Warsaw,and Banach knew all of them.

There is another interesting story concerning Banach’sdoctoral dissertation. The referees were Zylinski andSteinhaus. In October 1920, Steinhaus, who was mentoringBanach, wrote to the dean to inquire about the date ofBanach’s doctoral exam, for it had been four months sinceBanach had delivered his dissertation. The dean repliedthat everything was ready for the exam, but they wereawaiting the referees’ report (one of whom was Steinhaushimself!). Indeed, when the joint report from Steinhaus andZylinski arrived, the exam took place immediately. Banach

had submitted his dissertation on June 24, the report isdated October 30, and the exam in mathematics and phy-sics took place on November 3. Bearing in mind that in1920, October 30 fell on a Saturday, November 3 wastherefore a Wednesday, and November 1 (Monday) is apublic holiday in Poland, everything must indeed havebeen prepared for the exam. Banach passed this exam witha unanimous grade of “excellent” from all four examiners.

On December 11, 1920, Banach passed the exam inphilosophy (the examining board consisted of the twophilosophers Kazimierz Twardowski and MscisławWartenberg and the dean, Zygmunt Weyberg). Banach hadnow fulfilled all the requirements for being granted thePhD degree, and in many sources (including a CV signedby Banach; see [19]), 1920 is given as the year of Banach’sdoctorate. However, the precise rules for obtaining a PhDfrom Austro-Hungarian times (see [1]) had been retained byPoland after regaining its independence (see [14]).According to those rules, the candidate was allowed to callhimself “doctor” only after the doctoral conferment cere-mony, which in the case of Banach took place on January22, 1921. The official documents state that the academicianwho conferred the degree on Banach was KazimierzTwardowski. To a mathematician, that is surprising newsindeed. Why Twardowski, who was an eminent Polishphilosopher? What was his connection to Banach? Could hehave been his dissertation advisor? According to the rules

Figure 2. The main building of Lwow University in the 1920s. (Photo by A. Lenkiewicz, in the public domain.)

© 2021 The Author(s), Volume 43, Number 3, 2021 3

then in force, the conferment of a new doctorate had to becelebrated by a professor from the faculty appointed by thedean, and so there is no reason to regard Twardowski asthe supervisor of Banach’s thesis. By analogy, one mightincorrectly claim that Steinhaus’s supervisor in Gottingen in1911 was the German botanist Gustav Albert Peter, whoplayed the same role as Twardowski in Banach’s case (fordetails, see [9]).

It is frequently said that Banach was not a universitygraduate, so the fact that he obtained a position at thePolytechnic and a university doctorate was exceptional.This is also slightly misleading. According to the rules thatwere then in effect in Poland [14], four years of study at theuniversity was enough for one to be eligible for a PhD, buteven that requirement could be relaxed. The professors ofa faculty could, at their discretion, allow someone withoutstanding achievements to apply for a PhD. Moreover, inthose years, there was no precise definition of who countedas a university graduate. Banach had studied at the LvovPolytechnic for precisely four years, which was enough.

Let us add some further information on Banach’s doc-torate. His dissertation was truly impressive. In it, Banachdefined a space that came to be known as the Banachspace. He proved some results on such spaces, some the-orems on linear operators, and the theorem that is nowknown as the Banach fixed-point theorem. His dissertationwas published two years later as [2]. For an excellentanalysis of the results contained therein, see [5, 15, 18, 23].Banach’s dissertation was of fundamental importance forthe development of mathematics in the twentieth century,which, however, was not evident at the outset.

Independently of Banach, Norbert Wiener presented avery similar axiomatic system for vector spaces. It waspublished also in 1920, but in the fall, so Banach was a littlefaster out of the gate (see [35]). The name “Banach space”was first introduced by Maurice Frechet in [16] as les espacesde Banach. Earlier, Banach and his collaborators had usedthe name “B-spaces.” As was noted in [23], this term firstappeared in print in Steinhaus’s paper [31]. Steinhaus wrotethere about der B-Raum. For some time, the spaces inquestion were called “Banach–Wiener spaces.” In any case,Wiener wrote later in [35], “For a short while I kept pub-lishing a paper or two on this topic, but I gradually left thefield. At present these spaces are quite justly named afterBanach alone.”

Wiener abandoned this topic, but Banach continued hisresearch. Around 1930, Banach proved three theoremsfundamental for functional analysis: the Hahn–Banachtheorem, the Banach–Steinhaus theorem, and the Banachopen mapping theorem. This marked the real starting pointof the development of functional analysis.4 In 1931, Banachpublished the monograph Operacje liniowe (Linear Oper-ations), and a year later, he published the same book inFrench [3]. At that point, the mathematical world recog-nized the importance of Banach’s results. Banach wasinvited to deliver a plenary lecture at ICM 1936, in Oslo.

So, the story of the extraordinary circumstances ofBanach’s doctorate is completely false. However, it is saidthat “underneath gossip there’s a kernel of truth.” Let usthen dig further in an attempt to discover it.

This is a good place to recall the illustrious figure ofAndrzej Turowicz (1904–1989), a mathematician, priest,and monk active mostly in Krakow, but who also spentsome time working in Lvov; see [13]. Turowicz knew manyexcellent stories, abounding in colorful detail, aboutmathematics and mathematicians of his time. It was notunusual for participants in various meetings that he atten-ded to ask him to share some of his anecdotes. WheneverTurowicz had himself been a witness of an event, herecounted it with great accuracy, and one could be surethat things had really happened that way, but there werealso stories he had heard from others.

On November 17, 1984, the Jagiellonian University Stu-dents’ Mathematics Society (see [10]) invited severalmathematicians to share their memories during a specialmeeting. Their reminiscences were taped. Turowicz wasone of the guests. He contributed the anecdote aboutBanach’s PhD exam, beginning with the words, “This is astory I heard from Nikodym, and I am repeating it here atNikodym’s responsibility.” Turowicz recounted this eventon several occasions and always credited it to Nikodym.The same attribution is also given in [20].

It was Nikodym whose conversation with Banach wasaccidentally overheard by Steinhaus in Krakow. Later,Nikodym became a prominent mathematician; after WorldWar II he emigrated to the United States. More informationabout him can be found in [6] and [34].

And it turns out that it was Nikodym who was reluctantto obtain a PhD. He used to ask, “Will it make me anywiser?” In 1924, Nikodym (aged 35), still without a PhD,and his wife, Stanisława (who also was a mathematician),moved from Krakow to Warsaw. Walerian Piotrowski madea very solid investigation concerning PhDs in mathematicsat Warsaw University in the interwar period (see [24, 25]).According to [25], Wacław Sierpinski decided to take thematter of Nikodym’s PhD exam into his own hands. Heinvited Nikodym to a cafe and began to talk with him. Aftera while, the dean of the department “accidentally”appeared in the cafe and joined the conversation, whichquickly drifted toward mathematics. More than an hourlater, Sierpinski said to Nikodym, “Congratulations. Youhave just passed your PhD exam.”5

In our opinion, this is the source of the urban legendabout Banach’s doctorate. We will never know whetherNikodym gave Turowicz a twisted account of his own PhDexam, changing the main protagonist’s name in the pro-cess, or whether Turowicz missed something. Our view isthat the first explanation is more likely.

We conclude with yet another story about Banach thathas been widely circulated (see, for example, [20]). To wit,toward the end of the 1930s (around 1937), Wiener sup-posedly intended to offer Banach a job in the United States,and he sent an emissary to Lvov to convince Banach to

4For an excellent description of the origins of functional analysis, see [29].5Stanisława Nikodym obtained a PhD in Warsaw the same year, without any such stratagem. She was the first women in Poland to be awarded a PhD in mathematics.

4 THE MATHEMATICAL INTELLIGENCER

accept the offer. Banach asked, “How much is ProfessorWiener willing to pay?” The visitor from the United Statesanswered, “Ah, we anticipated this question” and handedBanach a check signed by Wiener on which only the digit 1was written. “Please add to it as many zeros as you deemfit!” Banach replied, “Such a sum is too small for me toleave Poland.”

This story cannot be true.6 For one thing, there was noreason for Wiener to invite Banach, as by then he had longceased to work in Banach’s field of interest. Moreover,neither Wiener nor Steinhaus mentions in his memoirs(respectively [35] and [32]) any plans by Wiener to inviteBanach to the United States. In his autobiography [35],Wiener does not write anything about Banach personally;he mentions Banach only on the occasion of writing aboutBanach/Banach–Wiener spaces. Also, Wiener’s putativeoffer appears financially implausible: someone faced withsuch an offer could well have written in thirty zeros ormore. Nevertheless, there exists a more direct argumentthat definitively falsifies this story. First, Banach spent theacademic year 1924/25 in Paris on a fellowship from theRockefeller Foundation. In the early 1930s, he applied forthat fellowship again, but his application was rejected,since “very few of the former fellows received a continu-ation” (see [28]). Apparently, Banach was not one of thosewhose fellowships were renewed. Second, in 1937–1938,Banach actually wanted to go to the United States! Not forgood, but for some time. Stanisław (Stan) Ulam (1909–1984), who had moved from Lvov to the United States in1935, was arranging a temporary position for Banach there.In [19] we find a letter from Banach to Ulam dated February14, 1938, in which he writes, “It seems from your letter thatmy visit to America is starting to become a reality .... Mostimportant, of course, is the amount they are willing to pay,

which should be adequate and suitable for a year’s stay forme and my wife and son .... Let me know what you wouldadvise me to prepare for my lectures in America.” Banach’svisit to the United States failed to materialize, probablybecause of the outbreak of World War II. Had Ulam man-aged to arrange it, Banach might have survived the war.

Is the story about the incredible salary completely ficti-tious? Or as in the case of the story about the doctorate, isthere a kernel of truth there?

Yes, there is! Enter the second of Banach’s friends fromhis Krakow years: Witold Wilkosz. Wilkosz and Banachattended the same secondary school in Krakow. Next,Wilkosz took up philological studies at the JagiellonianUniversity in Krakow. After two years, he switched tomathematics and left to study in Turin. In Italy, he waspreparing his doctoral dissertation, but due to the outbreakof the war he returned to Krakow. He received his PhDfrom the Jagiellonian University in 1918. He had a verybroad range of mathematical interests. Apart from obtain-ing important results in various branches of mathematics,he was a very active teacher and popularizer of science. It islittle known that Wilkosz was the author of the firstmonograph on topology in Poland [36] and the firstmonograph on topology by a Polish author publishedabroad [37].7 Wilkosz was also a pioneer in radio engi-neering and broadcasting in Poland.8 He constructed a newtype of radio receiver that came to be known as the“Wilkosz radio.” Being familiar with Wilkosz’s achieve-ments in this area, the famous Dutch electronics companyPhilips offered Wilkosz a job (see [27]). Philips offered hima salary of $5000 per month, which was then an incrediblylarge sum. To compare, the monthly salary of a professor atthe Jagiellonian University was equivalent to approximately$250. But Philips imposed a condition that everything thatWilkosz invented during his employment by the company

Figure 3. Stanisława and Otton Nikodym around 1924.

(Courtesy of the Polish Institute of Arts and Sciences of

America, Inc.)

Figure 4. Witold Wilkosz with his fultograph. (Courtesy of

Narodowe Archiwum Cyfrowe.)

6Or it could have originated as a joke made in the Scottish Cafe.7The book about the topology of the Euclidean plane and its subsets was published by Gauthier-Villars in 1931. It was volume 47 of the series Memorial des Sciences

Mathematiques published under the auspices of l’Academie des Sciences de Paris and ten other national academies. In it, Wilkosz presented an outline of the research

in this topic up to the end of the 1920s. Several open problems were also presented.8He also obtained some results concerning mathematical models of generating radio waves, especially connected to the van der Pol equation.

© 2021 The Author(s), Volume 43, Number 3, 2021 5

would become the property of Philips. Wilkosz did notaccept the offer and remained in Krakow.

Banach, Wilkosz, and Nikodym often discussed mathe-matics among themselves, but they never published a jointpaper. Nevertheless, in his famous dissertation [2], Banachannounced that he and Wilkosz were preparing a paper ona similar topic. Such a paper never appeared.

Thus two famous stories attributed to Banach really tookplace, although not precisely in the form in which they arecommonly narrated. In particular, they did not concernBanach, but his two colleagues who used to discussmathematical problems with him in their younger years:Otton Nikodym and Witold Wilkosz.

A good story is always welcome. But the truth is evenmore welcome.

Danuta CiesielskaLudwik and Aleksander Birkenmajer Institute for theHistory of SciencePolish Academy of SciencesNowy Swiat 72, 00-330 Warszawa, Polande-mail: smciesie@cyfronet.krakow.pl

Krzysztof CiesielskiMathematics Institute, Jagiellonian UniversityLojasiewicza 6, 30-348 Krakow, Polande-mail: krzysztof.ciesielski@im.uj.edu.pl

ACKNOWLEDGMENTS

We would like to thank Krzysztof Kwasniewicz, Prze-

mysław M. Zukowski, and Wojciech Słomczynski for their

help. We also thank the referee for helpful comments.

Open Access This article is licensed under a Creative CommonsAttribution 4.0 International License, which permits use, sharing,adaptation, distribution and reproduction in any medium or format,as long as you give appropriate credit to the original author(s) andthe source, provide a link to the Creative Commons license, andindicate if changes were made. The images or other third partymaterial in this article are included in the article’s Creative Com-mons license, unless indicated otherwise in a credit line to thematerial. If material is not included in the article’s Creative Com-mons license and your intended use is not permitted by statutoryregulation or exceeds the permitted use, you will need to obtainpermission directly from the copyright holder. To view a copy ofthis licence, visit http://creativecommons.org/licenses/by/4.0/.

REFERENCES

[1] Austro-Hungarian Ministry of Culture and Education. Verordnung

des Ministers fur Cultus und Unterricht, betreffend eine abgean-

derte Rigorosenordnung fur die philosophischen Facultaten der

Universitaten der im Reichsrathe vertretenen Konigreiche und

Lander, 16.03.1899.

[2] S. Banach. Sur les operations dans les ensembles abstraits et

leur application aux equations integrales. Fund. Math. 3 (1922),

133–181.

[3] S. Banach. Theorie des operations lineaires. Monografie

Matematyczne. Warszawa, 1932.

[4] R. E. Bellman. Eye of the Hurricane: An Autobiography. World

Scientific, 1984.

[5] M. Bernkopf. The development of function spaces with particular

reference to their origins in integral equation theory. Arch. History

Exact Sci. 3 (1966), 1–96.

[6] D. Ciesielska. Stanisława and Otton Nikodym. In Against All

Odds, edited by E. Kaufholz-Soldat and N. M. R. Oswald,

pp. 151–175. Springer, 2020.

[7] D. Ciesielska and K. Ciesielski. Banach and Nikodym on the

bench in Krakow again. European Mathematical Society

Newsletter 104 (2017), 25–29.

[8] D. Ciesielska and K. Ciesielski. Banach in Krakow: a case

reopened. Math. Intelligencer 35:3 (2013), 64–68.

[9] D. Ciesielska, L. Maligranda, and J. Zwierzynska. Doktoraty

Polakow w Getyndze. Matematyka. Analecta 28:2 (2019), 73–

116.

[10] K. Ciesielski. 100th anniversary of the Jagiellonian University

students’ Maths Society. Math. Intelligencer 17:4 (1995), 42–46.

[11] K. Ciesielski. Lost legends of Lvov 2: Banach’s grave. Math.

Intelligencer 10:1 (1988), 50–51.

[12] K. Ciesielski. On some details of Stefan Banach’s life. Opuscula

Math. 13 (1993), 71–74.

[13] K. Ciesielski and Z. Pogoda. Conversation with Andrzej Turowicz.

Math. Intelligencer 10:4 (1988), 13–20.

[14] T. Czezowski (editor). Zbior ustaw i rozporzadzen o studiach

uniwersyteckich oraz innych przepisow waznych dla studentow

uniwersytetu, ze szczegolnym uwzglednieniem Uniwersytetu

Stefana Batorego w Wilnie. Wilno, 1926.

[15] R. Duda. Pearls from a Lost City: The Lvov School of Mathe-

matics. American Mathematical Society, 2014.

[16] M. Frechet. Les espaces abstraits topologiquement affines. Acta

Math. 47 (1926), 25–52.

[17] L. Garding and L. Hormander. Why is there no Nobel Prize in

mathematics? Math. Intelligencer 7:3 (1985), 73–74.

[18] F. Jaeck. Generality and structures in functional analysis: the

influence of Stefan Banach. In The Oxford Handbook of

Generality in Mathematics and the Sciences, edited by K.

Chemla, R. Chorlay, and D. Rabouin, pp. 223–254. Oxford

University Press, 2016.

[19] E. Jakimowicz and A. Miranowicz (editors). Stefan Banach.

Remarkable Life, Brilliant Mathematics. Gdansk University Press,

2010.

[20] R. Kałuza. Through a Reporter’s Eyes: The life of Stefan Banach.

Birkhauser, 1996.

[21] S. G. Krantz. Mathematical Apocrypha Redux. Mathematical

Association of America, 2005.

[22] L. Maligranda. 100-lecie doktoratu Stefana Banacha. To appear

in Wiad. Mat. 56 (2020).

[23] L. Maligranda. ‘‘B’’jak przestrzen Banacha. Wiad. Mat. 56

(2020), 15–27.

[24] W. Piotrowski. Doktoraty z matematyki i logiki na Uniwersytecie

Warszawskim w latach 1915–1939. In Dzieje Matematyki Polskiej

II, edited by W. Wiesław, pp. 97–131. Instytut Matematyczny

Uniwersytetu Wrocławskiego, 2013.

[25] W. Piotrowski. Jeszcze w sprawie biografii Ottona i Stanisławy

Nikodymow. Wiad. Mat. 50 (2014), 69–74.

6 THE MATHEMATICAL INTELLIGENCER

[26] J. Prytuła. Doktoraty z matematyki i logiki na Uniwersytecie Jana

Kazimierza we Lwowie w latach 1920-1938. In Dzieje Matematyki

Polskiej, edited by W. Wiesław, pp. 137–161. Instytut Matem-

atyczny Uniwersytetu Wrocławskiego, 2012.

[27] K. Rakoczy-Pindor. Przygoda matematyka z radiem. Wiad. Mat.

39 (2003), 151–156.

[28] R. Siegmund-Schultze. Rockefeller and the Internationalization of

Mathematics Between the Two World Wars. Springer, 2001.

[29] R. Siegmund-Schultze. The origins of functional analysis. In A

History of Analysis, edited by N. Jahnke, pp. 385–407. American

Mathematical Society, 2003.

[30] W. Sierpinski. The Warsaw School of Mathematics and the

present state of mathematics in Poland. Polish Review 4 (1959),

51–63.

[31] H. Steinhaus. Anwendungen der Funktionalanalysis auf einige

Fragen der reellen Funktionentheorie. Studia Math. 1 (1929), 51–

81.

[32] H. Steinhaus. Mathematician for All Seasons: Recollections and

Notes, Vol. 1 (1887–1945). Birkhauser, 2015.

[33] H. Steinhaus. Stefan Banach. Wiad. Mat. 4 (1961), 251–259.

[34] W. Szymanski. Who was Otto Nikodym? Math. Intelligencer 12:2

(1990), 27–31.

[35] N. Wiener. I Am a Mathematician: The Later Life of a Prodigy. MIT

Press, 1956.

[36] W. Wilkosz. Podstawy ogolnej teorji mnogosci. Trzaska, Evert &

Michalski, Warszawa, 1925.

[37] W. Wilkosz. Les proprietes topologiques du plan euclidien.

Gauthier-Villars, 1931.

Publisher’s Note Springer Nature remains neutral with

regard to jurisdictional claims in published maps and

institutional affiliations.

© 2021 The Author(s), Volume 43, Number 3, 2021 7